Asymptotics for cliques in scale-free random graphs
Fraser Daly, Alastair Haig, Seva Shneer

TL;DR
This paper derives detailed asymptotic formulas for the expected number of cliques of all sizes in a scale-free inhomogeneous random graph model, highlighting differences between integer and non-integer degree exponents.
Contribution
It provides a comprehensive set of asymptotic results for clique counts in the Chung--Lu model with power-law weights, including non-integer exponents and partial results for integer exponents.
Findings
Complete asymptotics for all clique sizes with non-integer alpha
Explanation of differences in asymptotics for integer alpha
Partial results for integer alpha case
Abstract
In this paper we establish asymptotics (as the size of the graph grows to infinity) for the expected number of cliques in the Chung--Lu inhomogeneous random graph model in which vertices are assigned independent weights which have tail probabilities , where and is a slowly varying function. Each pair of vertices is connected by an edge with a probability proportional to the product of the weights of those vertices. We present a complete set of asymptotics for all clique sizes and for all non-integer . We also explain why the case of an integer is different, and present partial results for the asymptotics in that case.
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Taxonomy
TopicsStochastic processes and statistical mechanics · Limits and Structures in Graph Theory · Graph theory and applications
